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Chapter 3: Problem 58

For Exercises 58 and \(59,\) assume that \((a, b)\) is a point on the graph of \(y=f(x),\) and specify the corresponding point on the graph of each equation. [For example, the point that corre- sponds to \((a, b)\) on the graph of \(y=f(x-1)\) is (\(a+1, b).] (a) \)y=f(x-3)\( (b) \)y=f(x)-3\( (c) \)y=f(x-3)-3\( (d) \)y=-f(x)$

Short Answer

Expert verified
(a) (a+3, b); (b) (a, b-3); (c) (a+3, b-3); (d) (a, -b)

Step by step solution

01

Translation Right by 3 Units

The point \(a, b\) on the graph of \(y=f(x-3)\) corresponds to a horizontal translation of the graph of \(y=f(x)\) 3 units to the right. Therefore, the new point is \(a+3, b\).
02

Vertical Shift Down by 3 Units

For the graph equation \(y = f(x) - 3\), this represents a downward shift of the entire graph by 3 units. The corresponding point becomes \(a, b-3\).
03

Combination of Horizontal and Vertical Shift

The equation \(y=f(x-3)-3\) implies both a horizontal shift to the right by 3 units, and a vertical shift down by 3 units. Thus, the point \(a, b\) transforms into \(a+3, b-3\).
04

Reflection about the x-axis

The equation \(y=-f(x)\) represents a reflection of the graph about the x-axis. The point \(a, b\) will change to \(a, -b\) as all y-values are negated, flipping the graph upside down.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Translation
When working with functions, horizontal translation is a common transformation which involves sliding the entire graph left or right on the coordinate plane.
This type of transformation affects the x-values of the points on the graph without altering their y-values. For instance, given the point \(a, b\) on the graph of \ y = f(x) \, if we apply the transformation \( y = f(x - c) \), we are translating the graph horizontally by \(c\) units.
  • If \(c\) is positive, the graph shifts \(c\) units to the right.
  • If \(c\) is negative, the graph shifts \(|c|\) units to the left.
Thus, for the function \(y = f(x - 3)\), the point \((a, b)\) will move to \((a + 3, b)\).
This is because subtracting 3 in \(f(x-3)\) effectively tells us to shift the graph 3 units right.
Vertical Shift
Vertical shifts are transformations that involve moving the graph of a function up or down on the coordinate plane.
This transformation affects the y-values of the points in the graph, while the x-values remain unchanged. In the equation \(y = f(x) + c\), the entire graph shifts vertically.
  • If \(c\) is positive, the graph moves \(c\) units upward.
  • If \(c\) is negative, the graph moves \(|c|\) units downward.
Consider the function \(y = f(x) - 3\), the point \((a, b)\) shifts to \((a, b-3)\).
This process involves lowering each point on the graph by 3 units, thereby moving the entire graph downward.
Reflection
Reflection is a type of transformation that flips the graph over a specified axis. In the case of the equation \(y = -f(x)\), it involves reflecting the graph over the x-axis.
This means taking the y-values of each point on the graph and changing them to their negative counterparts. Hence, if a point on the original graph is \((a, b)\), it will transform to \((a, -b)\).
This flipping results in all points above the x-axis moving below it and vice-versa, effectively inverting the graph vertically. It's like looking at the graph in a mirror placed along the x-axis.
Graph of a Function
The graph of a function is a visual representation of all the points that satisfy the function's equation, showing the relationship between the input \(x\) and output \(y\).
Each point \((a, b)\) on the graph represents a specific pair of values that make the function true.
  • Transformations such as translations (both horizontal and vertical), reflections, and scaling can change the appearance and position of the graph.
  • These transformations do not alter the fundamental nature of the function but instead provide a new perspective on the same relationship.
Understanding function transformations allows you to manipulate and predict the behavior of a graph efficiently.
This is crucial in various math and science applications where functions model real-world scenarios.

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Most popular questions from this chapter

(a) A function \(f\) is said to be even if the equation \(f(-x)=f(x)\) is satisfied by all values of \(x\) in the domain of \(f .\) Explain why the graph of an even function must be symmetric about the \(y\) -axis. (b) Show that each function is even by computing \(f(-x)\) and then noting that \(f(x)\) and \(f(-x)\) are equal. (i) \(f(x)=x^{2}\) (iii) \(f(x)=3 x^{6}-\frac{4}{x^{2}}+1\) (ii) \(f(x)=2 x^{4}-6\)

Let \(P\) be a point with coordinates \((a, b),\) and assume that \(c\) and \(d\) are positive numbers. (The condition that \(c\) and \(d\) are positive isn't really necessary in this problem, but it will help you to visualize things.) (a) Translate the point \(P\) by \(c\) units in the \(x\) -direction to obtain a point \(Q,\) then translate \(Q\) by \(d\) units in the y-direction to obtain a point \(R\). What are the coordinates of the point \(R ?\) (b) Translate the point \(P\) by \(d\) units in the \(y\) -direction to obtain a point \(S,\) then translate \(S\) by \(c\) units in the \(x\) -direction to obtain a point \(T .\) What are the coordinates of the point \(T ?\) (c) Compare your answers for parts (a) and (b). What have you demonstrated? (Answer in complete sentences.)

Let \(f(x)=\sqrt{x^{3}+2 x+17}\) and \(g(x)=x+6\) (a) What is the relationship between the graphs of the two functions \(f\) and \(f \circ g ?\) (As in Exercise 21 , the idea here is to answer without looking at the graphs.) (b) Use a graphing utility to check your answer in part (a).

Let \(f(x)=-\frac{2 x+2}{x}\) (a) Find \(f[f(x)]\) (b) Use a graphing utility to graph \(y=f[f(x)] .\) Display the graph using true proportions. What type of symmetry does the graph appear to have? (c) The result in part (b) suggests that the inverse of the function \(f \circ f\) is again \(f \circ f .\) Use algebra to show that this is indeed correct.

Let \(f(x)=x^{2}\) and \(g(x)=2 x-1\). (a) Compute \(\frac{f[g(x)]-f[g(a)]}{g(x)-g(a)}\). (b) Compute \(\frac{f[g(x)]-f[g(a)]}{x-a}\).
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